The finite-dimensional decomposition property in non-Archimedean Banach spaces

A non-Archimedean Banach space has the orthogonal finite-dimensional decomposition property(OFDDP) if it is the orthogonal direct sum of a sequence of finite-dimensional subspaces.This property has an influence in the non-Archimedean Grothendieck's approximation theory,where an open problem is the following: Let E be a non-Archimedean Banach space of countable type with the OFDDP and let D be a closed subspace of E.Does D have the OFDDP? In this paper we give a negative answer to this question; we construct a Banach space of countable type with the OFDDP having a one-codimensional subspace without the OFDDP.Next we prove that,however,for certain classes of Banach spaces of countable type,the OFDDP is preserved by taking finite-codimensional subspaces.